uu27itu: f 22 wallpaper

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Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the three-dimensional crystallographic groups (also called space groups).

Wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns that are very different in style, color, scale or orientation may belong to the same group.

Sometimes two categorizations are meaningful, one based on shapes alone and one also including colors. When colors are ignored there may be more symmetry. In black and white there are also 17 wallpaper groups; e.g., a colored tiling is equivalent with one in black and white with the colors coded radially in a circularly symmetric "bar code" in the centre of mass of each tile.

All 17 groups were used by Egyptian craftsmen[citation needed], and used extensively in the Muslim world. A proof that there were only 17 possible patterns was first carried out by Evgraf Fedorov in 1891 and then derived independently by George Pólya in 1924.